What is the best way to store a list of objects with some linked data?
Custom properties.

Could store the segments in a custom property on the object  with the same name as the modifier.

for obj in self.stored_selection:
    for mod in obj.modifiers:            
        if mod.type == "BEVEL" and mod.segments &gt; 1:
            obj[mod.name] = mod.segments
            mod.segments = 1


and restore with

for obj in self.stored_selection:
    for mod in obj.modifiers:            
        if mod.type == "BEVEL":
            segments = obj.get(mod.name)
            if segments:
                mod.segments = segments
                #del(obj[mod.name]) # remove prop?


Another method would be to make a list of all the bevel modifiers with the segments value.  Need to proceed with a little caution when saving lists of object, or in this case modifier, references. If you remove an object that has the modifier, then try and reference later from list may produce a system fault and crash blender.

bevel_mods = [m, m.segments for o in context.selected_objects
        for m in o.modifiers
        if m.type == 'BEVEL'
        and m.segments &gt; 1]

for m, segments in bevel_mods:
    # print object name modifier name segments
    print(m.id_data.name, m.name, segments)